Question: $84$ people attended a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was $80$ less than $3$ times the number of away team fans. How many home team and away team fans attended the game?
Solution: Let $x$ equal the number of home team fans and $y$ equal the number of away team fans. The system of equations is then: ${x+y = 84}$ ${x = 3y-80}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${3y-80}$ for $x$ in the first equation. ${(3y-80)}{+ y = 84}$ Simplify and solve for $y$ $ 3y-80 + y = 84 $ $ 4y-80 = 84 $ $ 4y = 164 $ $ y = \dfrac{164}{4} $ ${y = 41}$ Now that you know ${y = 41}$ , plug it back into ${x = 3y-80}$ to find $x$ ${x = 3}{(41)}{ - 80}$ $x = 123 - 80$ ${x = 43}$ You can also plug ${y = 41}$ into ${x+y = 84}$ and get the same answer for $x$ ${x + }{(41)}{= 84}$ ${x = 43}$ There were $43$ home team fans and $41$ away team fans.